I can't see any reason to go out via the West Yellowstone entrance to go East. If not wanting to do the Cody Route, I'd use the North Exit by Mammoth to the interstate then East. US 89 to I-90.
We drove by
Pompey's Pillar not knowing what it was in 2009, about 25 miles east of Billings, Montana, on our way to Yellowstone. "It bears the only remaining physical evidence of the Lewis and Clark Expedition, which appears on the trail today as it did 200 years ago. On the face of the 150-foot butte, Captain William Clark carved his name on July 25, 1806, during his return to the United States through the beautiful Yellowstone Valley." Might be worth a stop, if time allows.
Cody Route is doable....some climbing then descent into Cody, then some`climbing & then descenting to get to the interstate by Sheridan.
If you do make it to Cody, try to see the Buffalo Bill Historical Center with Five Museums Under One Roof.
http://www.bbhc.org/These 4 were really great:
Buffalo Bill Museum
Whitney Gallery of Western Art
Plains Indian Museum
Cody Firearms Museum
The Cody Chamber of Commerce has a nice Visitor Center (full of information) & with a really nice art gallery.
http://codychamber.org/visitorcenter.htmlOn our 2nd trip to Yellowstone, we went up to Mt. Rushmore & Black Hills, then went to Yellowstone via Cody, WY. Stopped by Devil's Tower & detoured up to see the Little Big Horn Battlefield on the way to Cody. We Exited via Mammoth on our first trip. We enter that route in 2009 & exited via Cody. Not as much climbing as via Cody.
We've gone by Thermopolis, WY twice to do a soak in the hot spring water at the State Bath House. Very neat. We over nighted in the Wind River Canyon in Aug '09 & at one of the Boysen State Park campgrounds by the lake some years ago. But each time, we had already gone to Yellowstone, before stopping.
thermopolis.com/todo/hot-springs-state-park/This old post has a couple photos of the Fishing Bridge to Cody Route:
Clicky. Photos at the end of report.
As for which to see if can't do both...hard to say. As someone mentioned, you can do each of them in a somewhat drive by manner.
We did Devil's Tower that way the first time there. Did an overnight by the interstate & dove up gave it a look, then returned to the campground. We didn't stay a very long time at the Battlefield, but drove the loop & stopped to see the sights, read about each stop & visited the visitor center. Then on our way.
It might make more sense to go out the Mammoth route if going to the Battlefield. It would be on the way to the Tower, etc.
We drove up to the Battlefield, then returned to Sheridan.
Seems, if I am remembering correctly, one can go from Cody to Buffalo as well. Missing Sheridan completely, if not going to the Battlefield. I believe it is somewhat less climbing that route.